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PoincareSection

is a setting for the option TrekGenerator that specifies that the Poincaré section for differential equations is used to generate treks.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
See Also
Tech Notes
EquationTrekker`
EquationTrekker`

PoincareSection

PoincareSection

is a setting for the option TrekGenerator that specifies that the Poincaré section for differential equations is used to generate treks.

Details and Options

  • To use PoincareSection, you first need to load the Equation Trekker Package using Needs["EquationTrekker`"].
  • With TrekGenerator->{PoincareSection,method->value,…} possible method options are:
  • "SectionCondition"Nonewhen section variables will be shown
    "SectionVariables"Nonevariables to be plotted
    "FilterFunction"Nonefunction applied to variables before plotting
  • Options specific to NDSolve may also be included as method options.
  • The method options "SectionCondition" and "SectionVariables" are required to have a value other than None.
  • The method option "SectionCondition"->cond plots section variables whenever cond==0.
  • The setting for the method option "SectionVariables" should be a list of two of the dependent variables. If the differential equations are second order or higher, then derivatives of the dependent variables may also be chosen.
  • With "FilterFunction"->f, the argument supplied to f is a list of the coordinates to be plotted.

Examples

Basic Examples  (1)

Wolfram Language code: Needs["EquationTrekker`"]

This shows an example of treks generated by Duffing's equation:

Wolfram Language code: EquationTrekker[{ x''[t] + γ x'[ t] - x[t] + x[t] ^ 3 == ϵ Cos[ω t]}, x, {t, 0, 100}, PlotRange -> {{-2, 2}, {-2, 2}}, TrekParameters -> {γ -> .15, ϵ -> .3, ω -> 1.}]

Generating a Poincaré section for the same equation:

Wolfram Language code: EquationTrekker[{ x''[t] + γ x'[ t] - x[t] + x[t] ^ 3 == ϵ Cos[ω t]}, x, {t, 1000, 10000}, PlotRange -> {{-2, 2}, {-2, 2}}, TrekParameters -> {γ -> .15, ϵ -> .3, ω -> 1.}, TrekGenerator -> {PoincareSection, "SectionCondition" -> Mod[ω t, 2 π], "SectionVariables" -> {x, x'}, MaxSteps -> ∞}]